The Optimal Power Flow (OPF) problem for a power grid having alternating current (AC) circuits concerns the problem of determining bus voltages and generator power levels to minimize a cost function representing an operation of the power grid. The cost functions can include generator cost, resistive losses or tertiary voltage control. The minimization of the cost function is subject to OPF constrains that can include the AC power flow constraints, bounds on power generation, bounds on bus voltage magnitudes, bounds on thermal losses, and limits on power transfer on lines.
The conventional methods relax the OPF to find a solution using, e.g., the second order cone programming (SOCP). See, e.g., U.S. 2012/0150504. However, such approach provides optimal solution only under satisfaction of sufficient conditions for the relaxations. The sufficient conditions, e.g., rank condition, only hold under restrictive assumptions on the network topology and constraints on the OPF. Thus, the conventional methods are not suitable for analyzing OPF for arbitrarily structures of the power grid. In addition, the above methods provide no recourse when sufficient conditions for relaxation are not satisfied.
Thus, there remains a need to globally optimize electric power grids of various structures and configurations. In addition, when the power grid includes various storage devices, there is a need to optimize the power grid considering multiple time periods of optimization.